Letter reSeArCH
Methods
The North Atlantic region was chosen for this study partly because it is the world’s
busiest oceanic flight corridor. Owing to the zonally extended nature of the polar
jet stream in this region, transatlantic flights are typically affected by the strength
and position of the jet stream throughout their entire flight paths. The effects of
the jet stream on aircraft include headwinds, tailwinds and clear-air turbulence. A
further reason for choosing the North Atlantic is that—unlike the North Pacific—it
exhibits separate polar and subtropical jet streams, allowing an analysis of the polar
jet stream exclusively.
We used pressure-level zonal wind and temperature data from the ERA-Interim,
NCEP/NCAR and JRA-55 reanalysis datasets at six-hourly analysis intervals from
1 January 1979 to 31 December 2017 inclusive, giving 39 full years of data. All
datasets were used on a standard latitude–longitude grid (0.75° for ERA-Interim,
2.5° for NCEP/NCAR and 1.25° for JRA-55). Trends were calculated using ordi-
nary least-squares regression, and statistical significance was assessed at the 95%
confidence level (P < 0.05) according to a two-tailed t-test. The effect of temporal
autocorrelation on statistical significance was tested in the computed annual-mean
data and found to be negligible. Percentage changes were calculated using the
values of the fitted linear trend lines in 1979 and 2017.
To calculate the two-box zonal-mean bulk meridional temperature difference,
we first averaged the annual-mean temperature in a subtropical box (30°–50° N,
10°–80° W) and a subpolar box (50°–70° N, 10°–80° W), with a cosine(latitude)
weighting factor to account for the convergence of grid points at high latitudes.
The latitudinal bounds of these boxes were chosen to be approximately either side
of the climatological annual-mean jet stream latitude in the North Atlantic. We
then found the meridional temperature difference across the North Atlantic by
subtracting the subtropical box temperature from the subpolar box temperature.
The jet stream was analysed in the North Atlantic region (10°–80° W,
30°–70° N). The annual-mean regional-mean 250 hPa vertical shear in zonal wind
was calculated by taking a centred vertical finite difference using the annual-mean
zonal winds at 300 and 200 hPa:
−∂
∂≈u −
puu(200hPa) (300hPa)
100 hPa(3)
250 hPaWe also calculated trends in the annual-mean regional-mean (area-weighted) zonal
wind speed at 250 hPa over the North Atlantic region. Vertical profiles of vertical
shear trends were calculated by taking centred finite differences at intervals of
50 hPa for ERA-Interim and JRA-55, and from neighbouring pressure levels in
NCEP/NCAR (owing to the spacing of available pressure-level data).
The annual-mean regional-maximum vertical shear was calculated by a similar
centred-difference method: we first subtracted the zonal wind at 300 hPa from the
zonal wind at 200 hPa, and we then found the maximum value within the North
Atlantic region at each six-hourly interval, before averaging the maximum values
annually. For the annual-mean regional-maximum zonal wind speed, we found the
maximum zonal wind speed at 250 hPa within the North Atlantic region at each
six-hourly interval, before averaging annually. In both cases, the latitude at which
the maximum occurred was stored.
When the calculations in Fig. 3 are repeated using the annual-mean regional-
maximum vertical shear, instead of the annual-mean regional-mean vertical shear,
a statistically significant ensemble-mean increase of 11% (P < 0.01) in the shear
is found. The individual increases are 10% in ERA-Interim (P < 0.01), 18% in
NCEP/NCAR (P < 0.01), and 7% in JRA-55 (P < 0.01) (Extended Data Fig. 2).
These results confirm that the shear is strengthening within the jet core as well as
throughout the broader region influenced by the jet stream. The trends are not
attributable to a shift in the annual-mean latitude of the jet core, which shows no
statistically significant trend over the period (Extended Data Fig. 3).
We used the time derivative of the thermal wind balance equation to relate
linear trends in the meridional temperature gradient to linear trends in the vertical
wind shear. At 250 hPa, we calculated trends in the annual-mean values of ∂/up∂
(using the centred finite difference method outlined above) and ∂/Ty∂. The agree-
ment between the two was assessed through Pearson’s correlation coefficient using
an area-weighted pattern correlation.
According to thermal wind balance, the trend in the zonal wind speed in the
upper troposphere and lower stratosphere is given by the vertical integral of
equation ( 2 ). This vertical integral is performed throughout the depth of the free
troposphere, starting from the top of the planetary boundary layer. Temperature
gradients in the lower troposphere are included in the integral, and therefore Arctic
amplification at low levels is able to influence the wind speed at upper levels. For
example, written in equation form, we have:∫∫
∂
∂=∂
∂
∂
∂
+∂
∂
∂
∂
≈u
tR
fp tT
ypR
fp tT
yp(250hPa)
dd 0 (4)
p450 hPa 250 hPa0 450 hPawhere p 0 is the pressure at the top of the planetary boundary layer. Here, the free
troposphere has been divided into two layers at 450 hPa, by reference to Fig. 2.
The lower boundary term ∂/up() 0 ∂t arising from the vertical integration has
been neglected in equation ( 4 ), because the zonal wind speed in the lower
troposphere has no statistically significant trend in any of the reanalysis datasets,
as shown in Extended Data Fig. 1d–f. Our study shows that, on the right-hand
side of equation ( 4 ), the first integral (which includes the weakening low-level
temperature gradient from Arctic amplification) and the second integral (which
includes the strengthening upper-level temperature gradient) are essentially
equal and opposite when averaged over the North Atlantic region, thus largely
cancelling out and leaving no statistically significant trend in the upper-level
speed.Data availability
The NCEP/NCAR reanalysis data may be obtained from the National Oceanic
and Atmospheric Administration (NOAA) Oceanic and Atmospheric Research
(OAR) Earth System Research Laboratory (ESRL) Physical Sciences Division
(PSD), Boulder, Colorado, USA (https://www.esrl.noaa.gov/psd/). The ERA-
Interim and JRA-55 reanalysis data may be obtained from the Research Data
Archive at the National Center for Atmospheric Research (NCAR), Computational
and Information Systems Laboratory, Boulder, Colorado, USA (https://doi.org/
10.5065/D6CR5RD9 and https://doi.org/10.5065/D6HH6H41, respectively).Code availability
The analytical computer codes are publicly available at https://doi.org/
10.5281/zenodo.3238842.Acknowledgements S.H.L. acknowledges support through a PhD studentship
from the Natural Environment Research Council SCENARIO Doctoral Training
Partnership (reference NE/L002566/1).Author contributions S.H.L. and P.D.W. jointly conceived the study. S.H.L.
performed the data analysis and produced the figures with input from P.D.W.
and T.H.A.F. All authors contributed to writing the manuscript. The authors
discussed the results with each other at all stages.Competing interests The authors declare no competing interests.Additional information
Correspondence and requests for materials should be addressed to P.D.W.
Peer review information Nature thanks Darryn Waugh and Elizabeth A. Barnes
for their contribution to the peer review of this work.
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